65537th

The number 65537 is an integer that holds particular significance in mathematics, especially within number theory and constructible polygons. It is the fifth Fermat number, denoted as F5, which is defined by the formula 2^(2^n) + 1. For n=5, this yields 2^(2^5) + 1 = 2^32 + 1 = 4294967296 + 1 = 4294967297. However, the number referenced is 65537, which is F4, where n=4. This gives 2^(2^4) + 1 = 2^16 + 1 = 65536 + 1 = 65537.

Fermat numbers have a history tied to Pierre de Fermat, who conjectured that all numbers of this form are prime. While F0 through F4 (3, 5, 17, 257, and 65537) are indeed prime, this pattern does not continue. F5 was shown to be composite by Leonhard Euler in 1732, and subsequently, many other Fermat numbers have been proven to be composite.

The primality of 65537 is crucial because it is directly related to the constructibility of regular polygons using only a compass and straightedge. A regular n-gon can be constructed if and only if the number of sides n is the product of a power of 2 and any number of distinct Fermat primes. Since 65537 is a Fermat prime, a regular 65537-gon is constructible. This fact, discovered by Carl Friedrich Gauss, marked a significant advancement in classical geometry.